Fourier Series Of Continuous Functions Research Articles

Estimate of the Convergence Rate in the Riemann Localization Principle for Trigonometric Fourier Series of Continuous Functions

Estimate of the Convergence Rate in the Riemann Localization Principle for Trigonometric Fourier Series of Continuous Functions

On the constancy of signs and order of magnitude of Fourier–Haar coefficients

Abstract In the paper, we investigate the relation between the properties of functions and their Fourier–Haar coefficients. We show that for some classes of functions Fourier–Haar coefficients have constant signs and order of magnitude. In 1964, Golubov proved in [B. I. Golubov, On Fourier series of continuous functions with respect to a Haar system (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 1964, 1271–1296] that if f ⁢ ( x ) ∈ C ⁢ ( 0 , 1 ) {f(x)\in C(0,1)} , then its Fourier–Haar coefficients have constant signs when f ⁢ ( x ) {f(x)} is a nonincreasing function on [ 0 , 1 ] {[0,1]} , and in some cases those coefficients have a certain order of magnitude. In the present paper, we continue to investigate the properties of functions which follow from the behavior of their Fourier–Haar coefficients.

DIVERGENCE OF THE FOURIER SERIES OF CONTINUOUS FUNCTIONS WITH A RESTRICTION ON THE FRACTALITY OF THEIR GRAPHS

We consider certain classes of functions with a restriction on the fractality of their graphs. Modifying Lebesgue’s example, we construct continuous functions from these classes whose Fourier series diverge at one point, i.e. the Fourier series of continuous functions from this classes do not converge everywhere.

On the uniform convergence of negative order Cesaro means of Fourier series

The paper investigates the uniform convergence of negative order Cesaro means of Fourier series of continuous functions after correction of these functions on a set of small measure. We also give an example of continuous function, the (C,α) means of Fourier series of which diverge over any subsequence on the set of positive measure for any α∈(−1,−12).

О расходимости почти всюду рядов Фурье непрерывных функций двух переменных

О расходимости почти всюду рядов Фурье непрерывных функций двух переменных

On a class of summability methods for multiple Fourier series

The paper shows that the same properties which hold for the classical -means are preserved for a sufficiently large class of summability methods for multiple Fourier series involving rectangular partial sums. More precisely, Fourier series of continuous functions are uniformly summable by these methods, and Fourier series of functions from the class are summable almost everywhere.Bibliography: 6 titles.

A weak generalized localization of multiple Fourier series of continuous functions with a certain module of continuity

Let E be an arbitrary measurable set, E ⊂ T N = [−π, π)N, N ≥ 1, μE > 0, and let μ be a measure. In this paper, a weak generalized almost everywhere localization is studied, i.e., for given subsets E 1 ⊂ E, μE 1 > 0 we study the almost everywhere convergence of multiple trigonometric Fourier series of functions that are zero on E. We obtain sufficient conditions for the almost everywhere convergence of multiple Fourier series (summable over rectangles) of functions from {ie031-01}, as δ → 0 on E 1. These conditions are given in terms of the structure and geometry of the sets E 1 and E and are related to certain orthogonal projections of the sets; they are called the {ie031-02} property of the set E. Previously, one of the authors had introduced the {ie031-03}, k = 1, 2, properties of the set E, which are related to one-dimensional and two-dimensional projections of the sets E and E 1 respectively, as sufficient conditions for the almost everywhere convergence of Fourier series of functions from L 1(T N) and L p (T N), p > 1. The results presented generalize these ideas.

λ-Divergence of the Fourier Series of Continuous Functions of Several Variables

In this paper, we consider the behavior of rectangular partial sums of the Fourier series of continuous functions of several variables with respect to the trigonometric system. The Fourier series is called λ-convergent if the limit of rectangular partial sums over all indices \(\vec M = (M_1 ,...,M_n )\) for which \(1/\lambda \leqslant M_j /M_k \leqslant \lambda \) for all j and k exists. In the space of arbitrary even dimension 2m we construct an example of a continuous function with an estimate of the modulus of continuity \(\omega (F,\delta ) = \mathop O\limits_{\delta \to + 0} ({\text{ln}}^{ - m} (1/\delta ))\) such that its Fourier series is λ-divergent everywhere for any \(\lambda >1\).

Some linear summation methods for Fourier series

New non-traditional methods are given that ensure uniform convergence for summation of trigonometric Fourier series of continuous functions.

Divergence everywhere of the Fourier series of continuous functions of several variables

The Fourier series of a function of real variables is said to be -convergent at a point for if there exists the limit over all indices such that for all and . An example of a continuous function of variables with modulus of continuity is constructed such that the Fourier series of with respect to the trigonometric system -diverges everywhere for an arbitrary fixed .

On strong Carleman means of multiple trigonometric Fourier series

We prove the strong Carleman summability of the Fourier series of continuous functions on the m-dimensional torus, with partial sums constructed over polyhedra of a certain class.

Convergence of linear means of multiple fourier series of continuous functions

Convergence of linear means of multiple fourier series of continuous functions

ON DIVERGENT FOURIER SERIES IN ORTHOGONAL SYSTEMS

In this work there is constructed a complete orthonormal system with respect to which the Fourier series of piecewise continuous non-identically-zero functions diverge a.e. There is also constructed an orthonormal system with respect to which the Fourier series of continuous functions converge a.e., while the Fourier series of bounded functions having at least one discontinuity of the first kind diverge a. e. Bibliography: 10 items.

TRIGONOMETRIC FOURIER SERIES OF CONTINUOUS FUNCTIONS DIVERGING ON A GIVEN SET

In this work divergent trigonometric Fourier series of continuous functions are investigated. Certain types of sets are determined which are sets of divergence, unbounded divergence and bounded divergence for Fourier series of continuous functions. From these results it follows that there exist sets of bounded divergence having Hausdorff dimension 1 and also sets having positive α-capacity. Bibliography: 19 items.

The convergence of the double Fourier series of continuous functions

The convergence of the double Fourier series of continuous functions

On typical means of fourier series of continuous functions

Let f(x) be continuous in (-π, π) and periodic with period 2π and let , The aim of this paper is to prove the following theorem.

Divergence everywhere of rearranged Fourier series of continuous functions

A pair of conjugate continuous functions are constructed whose Fourier series can be rearranged to form an unboundedly divergent series.

Unbounded divergence of Fourier series of continuous functions

For any given set E ⊂ [0, 2π), of measure zero, a functionf(t) e C (0, 2π), is constructed whose Fourier series is unboundedly divergent on E. If E is closed, there is a functionϕ(t) e C (0, 2π), whose Fourier series diverges unboundedly on E and converges on [0, 2π)E.

Unconditional convergence almost everywhere of Fourier series of continuous functions in the Haar system

A continuous function is constructed whose Haar-Fourier series, after a definite rearrangement of its terms, diverges almost everywhere. A function is also constructed which has the maximum degree of smoothness in the sense that if its smoothness is increased its Haar-Fourier series becomes unconditionally convergent almost everywhere.

A singularity of the Fourier series of continuous functions

A singularity of the Fourier series of continuous functions